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Theorem ereldm 8333
Description: Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ereldm.1 (𝜑𝑅 Er 𝑋)
ereldm.2 (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)
Assertion
Ref Expression
ereldm (𝜑 → (𝐴𝑋𝐵𝑋))

Proof of Theorem ereldm
StepHypRef Expression
1 ereldm.2 . . . 4 (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)
21neeq1d 3073 . . 3 (𝜑 → ([𝐴]𝑅 ≠ ∅ ↔ [𝐵]𝑅 ≠ ∅))
3 ecdmn0 8332 . . 3 (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅)
4 ecdmn0 8332 . . 3 (𝐵 ∈ dom 𝑅 ↔ [𝐵]𝑅 ≠ ∅)
52, 3, 43bitr4g 317 . 2 (𝜑 → (𝐴 ∈ dom 𝑅𝐵 ∈ dom 𝑅))
6 ereldm.1 . . . 4 (𝜑𝑅 Er 𝑋)
7 erdm 8295 . . . 4 (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
86, 7syl 17 . . 3 (𝜑 → dom 𝑅 = 𝑋)
98eleq2d 2901 . 2 (𝜑 → (𝐴 ∈ dom 𝑅𝐴𝑋))
108eleq2d 2901 . 2 (𝜑 → (𝐵 ∈ dom 𝑅𝐵𝑋))
115, 9, 103bitr3d 312 1 (𝜑 → (𝐴𝑋𝐵𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2115  wne 3014  c0 4276  dom cdm 5542   Er wer 8282  [cec 8283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-xp 5548  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-er 8285  df-ec 8287
This theorem is referenced by:  erth  8334  brecop  8386  eceqoveq  8398
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